Download PHYS4210 Electromagnetic Theory Spring 2009 Midterm Exam #1

Name:
PHYS4210
Electromagnetic Theory
Midterm Exam #1
Spring 2009
Thursday, 12 Feb 2009
This exam has four questions and you are to work all of them. You must hand in your
paper by the end of class time (3:50pm) unless prior arrangements have already been made
with the instructor.
Note that not all of the problems are worth the same number of points.
You may use your textbook, course notes, or any other reference you may have other
than another human. You are welcome to use your calculator or computer, although the test
is designed so that these are not absolutely necessary.
Good luck!
Problem 1:
Problem 2:
Problem 3:
Problem 4:
Total:
Problem 1 (20 points). Given a region of space V and the surface S which bounds it,
I
S
f (r)n̂dA =
Z
∇f (r)dV
V
where f (r) is an arbitrary function of position, and n̂ is the normal vector pointing outwards
from surface S. Prove this theorem in two steps, as we’ve done in class and on homework.
a. (10 points) First work in an infinitesimally small region dV , a box with one corner at
(x, y, z), and with sides dx, dy, and dz each of which tend to zero. Explicitly evaluate the
surface integral over the six sides, and show that it equals ∇f (r)dV .
b. (10 points) Second, argue why we can build an arbitrary region V out of such infinitesimally small regions, and how this proves the theorem above.
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Problem 2 (25 points). A charge
+Q sits a distance d from a grounded,
infinite conducting plane. A second
charge −Q sits a distance d farther,
along a line that is perpendicular to
the plane.
Find the magnitude and direction of
the force on the charge +Q.
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Problem 3 (25 points). See Schwartz
Problem 2-6. Consider the plane interface between a region I with dielectric
constant ε = 1.3 and a region II with
dielectric constant ε = 1.6 The electric
field E in region I is at 45◦ to the interface.
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Find the magnitude and direction of the
electric field in region II.
Problem 4 (30 points). A large conducting plate lies with one face in the (x, y) plane,
and vacuum for z > 0. This surface has a constant surface charge density σ.
a. (10 points) Starting from ∇ · E = 4πρ, i.e. Gauss’ Law, find the electric field vector E at
the surface by integrating Gauss’ Law, then using the divergence theorem, and then forming
the appropriate Gaussian surface. Write E in terms of the appropriate unit vector(s).
b. (10 points) Construct the electrostatic stress tensor T for this electric field. You may
write T either as a matrix in coordinates (x, y, z) or as a direct product of vectors and the
identity tensor 1.
c. (10 points) Show that the force per unit area on the surface of the conductor is 2πσ 2 .
Hint: You can use the same Gaussian surface as in part (a).